I am reading the book Complex Analysis by Gamelin and I am having trouble understanding the following definition:
Let $p_j$ be a sequence of non zero complex numbers. We say that $\prod p_j$ converges if $p_j\to 1$ and $\sum \log{p_j}$ converges (Here $\log$ denotes de principal branch of the logarithm).
I don't understand why the expression $\sum \log{p_j}$ makes sense because the $p_j$ can be negative integers, say $p_1=-1$, in this case how we define $\log(-1)$? (the principal branch of the logarithm is not defined for negative integers)